Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, X1, X2]
a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_FROM}(X) -> MARK(X)
A_{_HEAD}(cons(X, XS)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> A_{_HEAD}(mark(XS))
A_{_2ND}(cons(X, XS)) -> MARK(XS)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
MARK(from(X)) -> A_{_FROM}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(head(X)) -> A_{_HEAD}(mark(X))
MARK(head(X)) -> MARK(X)
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(2nd(X)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X1)
MARK(take(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(sel(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(2nd(X)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> MARK(XS)
A_{_2ND}(cons(X, XS)) -> A_{_HEAD}(mark(XS))
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(head(X)) -> MARK(X)
A_{_HEAD}(cons(X, XS)) -> MARK(X)
MARK(head(X)) -> A_{_HEAD}(mark(X))
MARK(from(X)) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_FROM}(X) -> MARK(X)

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A_{_2ND}(cons(X, XS)) -> A_{_HEAD}(mark(XS))

nine new Dependency Pairs are created:

A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
A_{_2ND}(cons(X, s(X''))) -> A_{_HEAD}(s(mark(X'')))
A_{_2ND}(cons(X, 0)) -> A_{_HEAD}(0)
A_{_2ND}(cons(X, nil)) -> A_{_HEAD}(nil)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(2nd(X)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> MARK(XS)
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(head(X)) -> MARK(X)
A_{_HEAD}(cons(X, XS)) -> MARK(X)
MARK(head(X)) -> A_{_HEAD}(mark(X))
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A_{_SEL}(s(N), cons(X, XS)) -> A_{_SEL}(mark(N), mark(XS))

18 new Dependency Pairs are created:

A_{_SEL}(s(from(X'')), cons(X, XS)) -> A_{_SEL}(a_{_from}(mark(X'')), mark(XS))
A_{_SEL}(s(head(X'')), cons(X, XS)) -> A_{_SEL}(a_{_head}(mark(X'')), mark(XS))
A_{_SEL}(s(2nd(X'')), cons(X, XS)) -> A_{_SEL}(a_{_2nd}(mark(X'')), mark(XS))
A_{_SEL}(s(take(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_take}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(sel(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(cons(X1', X2')), cons(X, XS)) -> A_{_SEL}(cons(mark(X1'), X2'), mark(XS))
A_{_SEL}(s(s(X'')), cons(X, XS)) -> A_{_SEL}(s(mark(X'')), mark(XS))
A_{_SEL}(s(0), cons(X, XS)) -> A_{_SEL}(0, mark(XS))
A_{_SEL}(s(nil), cons(X, XS)) -> A_{_SEL}(nil, mark(XS))
A_{_SEL}(s(N), cons(X, from(X''))) -> A_{_SEL}(mark(N), a_{_from}(mark(X'')))
A_{_SEL}(s(N), cons(X, head(X''))) -> A_{_SEL}(mark(N), a_{_head}(mark(X'')))
A_{_SEL}(s(N), cons(X, 2nd(X''))) -> A_{_SEL}(mark(N), a_{_2nd}(mark(X'')))
A_{_SEL}(s(N), cons(X, take(X1', X2'))) -> A_{_SEL}(mark(N), a_{_take}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, sel(X1', X2'))) -> A_{_SEL}(mark(N), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, cons(X1', X2'))) -> A_{_SEL}(mark(N), cons(mark(X1'), X2'))
A_{_SEL}(s(N), cons(X, s(X''))) -> A_{_SEL}(mark(N), s(mark(X'')))
A_{_SEL}(s(N), cons(X, 0)) -> A_{_SEL}(mark(N), 0)
A_{_SEL}(s(N), cons(X, nil)) -> A_{_SEL}(mark(N), nil)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
A_{_SEL}(s(N), cons(X, cons(X1', X2'))) -> A_{_SEL}(mark(N), cons(mark(X1'), X2'))
A_{_SEL}(s(N), cons(X, sel(X1', X2'))) -> A_{_SEL}(mark(N), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, take(X1', X2'))) -> A_{_SEL}(mark(N), a_{_take}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, 2nd(X''))) -> A_{_SEL}(mark(N), a_{_2nd}(mark(X'')))
A_{_SEL}(s(N), cons(X, head(X''))) -> A_{_SEL}(mark(N), a_{_head}(mark(X'')))
A_{_SEL}(s(N), cons(X, from(X''))) -> A_{_SEL}(mark(N), a_{_from}(mark(X'')))
A_{_SEL}(s(0), cons(X, XS)) -> A_{_SEL}(0, mark(XS))
A_{_SEL}(s(s(X'')), cons(X, XS)) -> A_{_SEL}(s(mark(X'')), mark(XS))
A_{_SEL}(s(sel(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(take(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_take}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(2nd(X'')), cons(X, XS)) -> A_{_SEL}(a_{_2nd}(mark(X'')), mark(XS))
A_{_SEL}(s(head(X'')), cons(X, XS)) -> A_{_SEL}(a_{_head}(mark(X'')), mark(XS))
A_{_SEL}(s(from(X'')), cons(X, XS)) -> A_{_SEL}(a_{_from}(mark(X'')), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(2nd(X)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> MARK(XS)
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(head(X)) -> MARK(X)
MARK(head(X)) -> A_{_HEAD}(mark(X))
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_HEAD}(cons(X, XS)) -> MARK(X)
A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(head(X)) -> A_{_HEAD}(mark(X))

nine new Dependency Pairs are created:

MARK(head(from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(head(head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
MARK(head(2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
MARK(head(take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
MARK(head(sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
MARK(head(cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
MARK(head(s(X''))) -> A_{_HEAD}(s(mark(X'')))
MARK(head(0)) -> A_{_HEAD}(0)
MARK(head(nil)) -> A_{_HEAD}(nil)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
A_{_SEL}(s(N), cons(X, cons(X1', X2'))) -> A_{_SEL}(mark(N), cons(mark(X1'), X2'))
A_{_SEL}(s(N), cons(X, sel(X1', X2'))) -> A_{_SEL}(mark(N), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, take(X1', X2'))) -> A_{_SEL}(mark(N), a_{_take}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, 2nd(X''))) -> A_{_SEL}(mark(N), a_{_2nd}(mark(X'')))
A_{_SEL}(s(N), cons(X, head(X''))) -> A_{_SEL}(mark(N), a_{_head}(mark(X'')))
A_{_SEL}(s(N), cons(X, from(X''))) -> A_{_SEL}(mark(N), a_{_from}(mark(X'')))
A_{_SEL}(s(0), cons(X, XS)) -> A_{_SEL}(0, mark(XS))
A_{_SEL}(s(s(X'')), cons(X, XS)) -> A_{_SEL}(s(mark(X'')), mark(XS))
A_{_SEL}(s(sel(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(take(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_take}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(2nd(X'')), cons(X, XS)) -> A_{_SEL}(a_{_2nd}(mark(X'')), mark(XS))
A_{_SEL}(s(head(X'')), cons(X, XS)) -> A_{_SEL}(a_{_head}(mark(X'')), mark(XS))
A_{_SEL}(s(from(X'')), cons(X, XS)) -> A_{_SEL}(a_{_from}(mark(X'')), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
MARK(head(cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
MARK(head(sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
MARK(head(take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
MARK(head(2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
MARK(head(head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
MARK(head(from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(2nd(X)) -> MARK(X)
A_{_2ND}(cons(X, XS)) -> MARK(XS)
MARK(2nd(X)) -> A_{_2ND}(mark(X))
MARK(head(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_HEAD}(cons(X, XS)) -> MARK(X)
A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(2nd(X)) -> A_{_2ND}(mark(X))

nine new Dependency Pairs are created:

MARK(2nd(from(X''))) -> A_{_2ND}(a_{_from}(mark(X'')))
MARK(2nd(head(X''))) -> A_{_2ND}(a_{_head}(mark(X'')))
MARK(2nd(2nd(X''))) -> A_{_2ND}(a_{_2nd}(mark(X'')))
MARK(2nd(take(X1', X2'))) -> A_{_2ND}(a_{_take}(mark(X1'), mark(X2')))
MARK(2nd(sel(X1', X2'))) -> A_{_2ND}(a_{_sel}(mark(X1'), mark(X2')))
MARK(2nd(cons(X1', X2'))) -> A_{_2ND}(cons(mark(X1'), X2'))
MARK(2nd(s(X''))) -> A_{_2ND}(s(mark(X'')))
MARK(2nd(0)) -> A_{_2ND}(0)
MARK(2nd(nil)) -> A_{_2ND}(nil)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Narrowing Transformation

Dependency Pairs:

A_{_SEL}(s(N), cons(X, cons(X1', X2'))) -> A_{_SEL}(mark(N), cons(mark(X1'), X2'))
A_{_SEL}(s(N), cons(X, sel(X1', X2'))) -> A_{_SEL}(mark(N), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, take(X1', X2'))) -> A_{_SEL}(mark(N), a_{_take}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, 2nd(X''))) -> A_{_SEL}(mark(N), a_{_2nd}(mark(X'')))
A_{_SEL}(s(N), cons(X, head(X''))) -> A_{_SEL}(mark(N), a_{_head}(mark(X'')))
A_{_SEL}(s(N), cons(X, from(X''))) -> A_{_SEL}(mark(N), a_{_from}(mark(X'')))
A_{_SEL}(s(0), cons(X, XS)) -> A_{_SEL}(0, mark(XS))
A_{_SEL}(s(s(X'')), cons(X, XS)) -> A_{_SEL}(s(mark(X'')), mark(XS))
A_{_SEL}(s(sel(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(take(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_take}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(2nd(X'')), cons(X, XS)) -> A_{_SEL}(a_{_2nd}(mark(X'')), mark(XS))
A_{_SEL}(s(head(X'')), cons(X, XS)) -> A_{_SEL}(a_{_head}(mark(X'')), mark(XS))
A_{_SEL}(s(from(X'')), cons(X, XS)) -> A_{_SEL}(a_{_from}(mark(X'')), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
MARK(2nd(cons(X1', X2'))) -> A_{_2ND}(cons(mark(X1'), X2'))
MARK(2nd(sel(X1', X2'))) -> A_{_2ND}(a_{_sel}(mark(X1'), mark(X2')))
MARK(2nd(take(X1', X2'))) -> A_{_2ND}(a_{_take}(mark(X1'), mark(X2')))
MARK(2nd(2nd(X''))) -> A_{_2ND}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(2nd(head(X''))) -> A_{_2ND}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, XS)) -> MARK(XS)
MARK(2nd(from(X''))) -> A_{_2ND}(a_{_from}(mark(X'')))
MARK(head(cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
MARK(head(sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
MARK(head(take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
MARK(head(2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
MARK(head(head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
MARK(head(from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))
MARK(2nd(X)) -> MARK(X)
MARK(head(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_HEAD}(cons(X, XS)) -> MARK(X)
A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(take(X1, X2)) -> A_{_TAKE}(mark(X1), mark(X2))

18 new Dependency Pairs are created:

MARK(take(from(X'), X2)) -> A_{_TAKE}(a_{_from}(mark(X')), mark(X2))
MARK(take(head(X'), X2)) -> A_{_TAKE}(a_{_head}(mark(X')), mark(X2))
MARK(take(2nd(X'), X2)) -> A_{_TAKE}(a_{_2nd}(mark(X')), mark(X2))
MARK(take(take(X1'', X2''), X2)) -> A_{_TAKE}(a_{_take}(mark(X1''), mark(X2'')), mark(X2))
MARK(take(sel(X1'', X2''), X2)) -> A_{_TAKE}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(take(cons(X1'', X2''), X2)) -> A_{_TAKE}(cons(mark(X1''), X2''), mark(X2))
MARK(take(s(X'), X2)) -> A_{_TAKE}(s(mark(X')), mark(X2))
MARK(take(0, X2)) -> A_{_TAKE}(0, mark(X2))
MARK(take(nil, X2)) -> A_{_TAKE}(nil, mark(X2))
MARK(take(X1, from(X'))) -> A_{_TAKE}(mark(X1), a_{_from}(mark(X')))
MARK(take(X1, head(X'))) -> A_{_TAKE}(mark(X1), a_{_head}(mark(X')))
MARK(take(X1, 2nd(X'))) -> A_{_TAKE}(mark(X1), a_{_2nd}(mark(X')))
MARK(take(X1, take(X1'', X2''))) -> A_{_TAKE}(mark(X1), a_{_take}(mark(X1''), mark(X2'')))
MARK(take(X1, sel(X1'', X2''))) -> A_{_TAKE}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(take(X1, cons(X1'', X2''))) -> A_{_TAKE}(mark(X1), cons(mark(X1''), X2''))
MARK(take(X1, s(X'))) -> A_{_TAKE}(mark(X1), s(mark(X')))
MARK(take(X1, 0)) -> A_{_TAKE}(mark(X1), 0)
MARK(take(X1, nil)) -> A_{_TAKE}(mark(X1), nil)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

A_{_SEL}(s(N), cons(X, sel(X1', X2'))) -> A_{_SEL}(mark(N), a_{_sel}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, take(X1', X2'))) -> A_{_SEL}(mark(N), a_{_take}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, 2nd(X''))) -> A_{_SEL}(mark(N), a_{_2nd}(mark(X'')))
A_{_SEL}(s(N), cons(X, head(X''))) -> A_{_SEL}(mark(N), a_{_head}(mark(X'')))
A_{_SEL}(s(N), cons(X, from(X''))) -> A_{_SEL}(mark(N), a_{_from}(mark(X'')))
A_{_SEL}(s(0), cons(X, XS)) -> A_{_SEL}(0, mark(XS))
A_{_SEL}(s(s(X'')), cons(X, XS)) -> A_{_SEL}(s(mark(X'')), mark(XS))
A_{_SEL}(s(sel(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(take(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_take}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(2nd(X'')), cons(X, XS)) -> A_{_SEL}(a_{_2nd}(mark(X'')), mark(XS))
A_{_SEL}(s(head(X'')), cons(X, XS)) -> A_{_SEL}(a_{_head}(mark(X'')), mark(XS))
A_{_SEL}(s(from(X'')), cons(X, XS)) -> A_{_SEL}(a_{_from}(mark(X'')), mark(XS))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
MARK(take(X1, cons(X1'', X2''))) -> A_{_TAKE}(mark(X1), cons(mark(X1''), X2''))
MARK(take(X1, sel(X1'', X2''))) -> A_{_TAKE}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(take(X1, take(X1'', X2''))) -> A_{_TAKE}(mark(X1), a_{_take}(mark(X1''), mark(X2'')))
MARK(take(X1, 2nd(X'))) -> A_{_TAKE}(mark(X1), a_{_2nd}(mark(X')))
MARK(take(X1, head(X'))) -> A_{_TAKE}(mark(X1), a_{_head}(mark(X')))
MARK(take(X1, from(X'))) -> A_{_TAKE}(mark(X1), a_{_from}(mark(X')))
MARK(take(s(X'), X2)) -> A_{_TAKE}(s(mark(X')), mark(X2))
MARK(take(sel(X1'', X2''), X2)) -> A_{_TAKE}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(take(take(X1'', X2''), X2)) -> A_{_TAKE}(a_{_take}(mark(X1''), mark(X2'')), mark(X2))
MARK(take(2nd(X'), X2)) -> A_{_TAKE}(a_{_2nd}(mark(X')), mark(X2))
MARK(take(head(X'), X2)) -> A_{_TAKE}(a_{_head}(mark(X')), mark(X2))
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(from(X'), X2)) -> A_{_TAKE}(a_{_from}(mark(X')), mark(X2))
MARK(2nd(cons(X1', X2'))) -> A_{_2ND}(cons(mark(X1'), X2'))
MARK(2nd(sel(X1', X2'))) -> A_{_2ND}(a_{_sel}(mark(X1'), mark(X2')))
MARK(2nd(take(X1', X2'))) -> A_{_2ND}(a_{_take}(mark(X1'), mark(X2')))
MARK(2nd(2nd(X''))) -> A_{_2ND}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(2nd(head(X''))) -> A_{_2ND}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, XS)) -> MARK(XS)
MARK(2nd(from(X''))) -> A_{_2ND}(a_{_from}(mark(X'')))
MARK(head(cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
MARK(head(sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
MARK(head(take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
MARK(head(2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
MARK(head(head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_HEAD}(cons(X, XS)) -> MARK(X)
MARK(head(from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
MARK(2nd(X)) -> MARK(X)
MARK(head(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_SEL}(s(N), cons(X, cons(X1', X2'))) -> A_{_SEL}(mark(N), cons(mark(X1'), X2'))

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MARK(sel(X1, X2)) -> A_{_SEL}(mark(X1), mark(X2))

18 new Dependency Pairs are created:

MARK(sel(from(X'), X2)) -> A_{_SEL}(a_{_from}(mark(X')), mark(X2))
MARK(sel(head(X'), X2)) -> A_{_SEL}(a_{_head}(mark(X')), mark(X2))
MARK(sel(2nd(X'), X2)) -> A_{_SEL}(a_{_2nd}(mark(X')), mark(X2))
MARK(sel(take(X1'', X2''), X2)) -> A_{_SEL}(a_{_take}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(sel(X1'', X2''), X2)) -> A_{_SEL}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(cons(X1'', X2''), X2)) -> A_{_SEL}(cons(mark(X1''), X2''), mark(X2))
MARK(sel(s(X'), X2)) -> A_{_SEL}(s(mark(X')), mark(X2))
MARK(sel(0, X2)) -> A_{_SEL}(0, mark(X2))
MARK(sel(nil, X2)) -> A_{_SEL}(nil, mark(X2))
MARK(sel(X1, from(X'))) -> A_{_SEL}(mark(X1), a_{_from}(mark(X')))
MARK(sel(X1, head(X'))) -> A_{_SEL}(mark(X1), a_{_head}(mark(X')))
MARK(sel(X1, 2nd(X'))) -> A_{_SEL}(mark(X1), a_{_2nd}(mark(X')))
MARK(sel(X1, take(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_take}(mark(X1''), mark(X2'')))
MARK(sel(X1, sel(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(sel(X1, cons(X1'', X2''))) -> A_{_SEL}(mark(X1), cons(mark(X1''), X2''))
MARK(sel(X1, s(X'))) -> A_{_SEL}(mark(X1), s(mark(X')))
MARK(sel(X1, 0)) -> A_{_SEL}(mark(X1), 0)
MARK(sel(X1, nil)) -> A_{_SEL}(mark(X1), nil)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

MARK(sel(X1, cons(X1'', X2''))) -> A_{_SEL}(mark(X1), cons(mark(X1''), X2''))
MARK(sel(X1, sel(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(sel(X1, take(X1'', X2''))) -> A_{_SEL}(mark(X1), a_{_take}(mark(X1''), mark(X2'')))
MARK(sel(X1, 2nd(X'))) -> A_{_SEL}(mark(X1), a_{_2nd}(mark(X')))
MARK(sel(X1, head(X'))) -> A_{_SEL}(mark(X1), a_{_head}(mark(X')))
MARK(sel(X1, from(X'))) -> A_{_SEL}(mark(X1), a_{_from}(mark(X')))
MARK(sel(0, X2)) -> A_{_SEL}(0, mark(X2))
MARK(sel(s(X'), X2)) -> A_{_SEL}(s(mark(X')), mark(X2))
MARK(sel(sel(X1'', X2''), X2)) -> A_{_SEL}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(sel(take(X1'', X2''), X2)) -> A_{_SEL}(a_{_take}(mark(X1''), mark(X2'')), mark(X2))
A_{_SEL}(s(N), cons(X, cons(X1', X2'))) -> A_{_SEL}(mark(N), cons(mark(X1'), X2'))
A_{_SEL}(s(N), cons(X, take(X1', X2'))) -> A_{_SEL}(mark(N), a_{_take}(mark(X1'), mark(X2')))
A_{_SEL}(s(N), cons(X, 2nd(X''))) -> A_{_SEL}(mark(N), a_{_2nd}(mark(X'')))
A_{_SEL}(s(N), cons(X, head(X''))) -> A_{_SEL}(mark(N), a_{_head}(mark(X'')))
A_{_SEL}(s(N), cons(X, from(X''))) -> A_{_SEL}(mark(N), a_{_from}(mark(X'')))
A_{_SEL}(s(0), cons(X, XS)) -> A_{_SEL}(0, mark(XS))
A_{_SEL}(s(s(X'')), cons(X, XS)) -> A_{_SEL}(s(mark(X'')), mark(XS))
A_{_SEL}(s(sel(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_sel}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(take(X1', X2')), cons(X, XS)) -> A_{_SEL}(a_{_take}(mark(X1'), mark(X2')), mark(XS))
A_{_SEL}(s(2nd(X'')), cons(X, XS)) -> A_{_SEL}(a_{_2nd}(mark(X'')), mark(XS))
A_{_SEL}(s(head(X'')), cons(X, XS)) -> A_{_SEL}(a_{_head}(mark(X'')), mark(XS))
A_{_SEL}(s(from(X'')), cons(X, XS)) -> A_{_SEL}(a_{_from}(mark(X'')), mark(XS))
MARK(sel(2nd(X'), X2)) -> A_{_SEL}(a_{_2nd}(mark(X')), mark(X2))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(XS)
MARK(sel(head(X'), X2)) -> A_{_SEL}(a_{_head}(mark(X')), mark(X2))
A_{_SEL}(s(N), cons(X, XS)) -> MARK(N)
MARK(sel(from(X'), X2)) -> A_{_SEL}(a_{_from}(mark(X')), mark(X2))
MARK(take(X1, cons(X1'', X2''))) -> A_{_TAKE}(mark(X1), cons(mark(X1''), X2''))
MARK(take(X1, sel(X1'', X2''))) -> A_{_TAKE}(mark(X1), a_{_sel}(mark(X1''), mark(X2'')))
MARK(take(X1, take(X1'', X2''))) -> A_{_TAKE}(mark(X1), a_{_take}(mark(X1''), mark(X2'')))
MARK(take(X1, 2nd(X'))) -> A_{_TAKE}(mark(X1), a_{_2nd}(mark(X')))
MARK(take(X1, head(X'))) -> A_{_TAKE}(mark(X1), a_{_head}(mark(X')))
MARK(take(X1, from(X'))) -> A_{_TAKE}(mark(X1), a_{_from}(mark(X')))
MARK(take(s(X'), X2)) -> A_{_TAKE}(s(mark(X')), mark(X2))
MARK(take(sel(X1'', X2''), X2)) -> A_{_TAKE}(a_{_sel}(mark(X1''), mark(X2'')), mark(X2))
MARK(take(take(X1'', X2''), X2)) -> A_{_TAKE}(a_{_take}(mark(X1''), mark(X2'')), mark(X2))
MARK(take(2nd(X'), X2)) -> A_{_TAKE}(a_{_2nd}(mark(X')), mark(X2))
MARK(take(head(X'), X2)) -> A_{_TAKE}(a_{_head}(mark(X')), mark(X2))
A_{_TAKE}(s(N), cons(X, XS)) -> MARK(X)
MARK(take(from(X'), X2)) -> A_{_TAKE}(a_{_from}(mark(X')), mark(X2))
MARK(2nd(cons(X1', X2'))) -> A_{_2ND}(cons(mark(X1'), X2'))
MARK(2nd(sel(X1', X2'))) -> A_{_2ND}(a_{_sel}(mark(X1'), mark(X2')))
MARK(2nd(take(X1', X2'))) -> A_{_2ND}(a_{_take}(mark(X1'), mark(X2')))
MARK(2nd(2nd(X''))) -> A_{_2ND}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
A_{_2ND}(cons(X, sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
A_{_2ND}(cons(X, 2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
A_{_2ND}(cons(X, head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(2nd(head(X''))) -> A_{_2ND}(a_{_head}(mark(X'')))
A_{_2ND}(cons(X, XS)) -> MARK(XS)
MARK(2nd(from(X''))) -> A_{_2ND}(a_{_from}(mark(X'')))
MARK(head(cons(X1', X2'))) -> A_{_HEAD}(cons(mark(X1'), X2'))
MARK(head(sel(X1', X2'))) -> A_{_HEAD}(a_{_sel}(mark(X1'), mark(X2')))
MARK(head(take(X1', X2'))) -> A_{_HEAD}(a_{_take}(mark(X1'), mark(X2')))
MARK(head(2nd(X''))) -> A_{_HEAD}(a_{_2nd}(mark(X'')))
MARK(head(head(X''))) -> A_{_HEAD}(a_{_head}(mark(X'')))
A_{_HEAD}(cons(X, XS)) -> MARK(X)
MARK(head(from(X''))) -> A_{_HEAD}(a_{_from}(mark(X'')))
MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sel(X1, X2)) -> MARK(X2)
MARK(sel(X1, X2)) -> MARK(X1)
MARK(take(X1, X2)) -> MARK(X2)
MARK(take(X1, X2)) -> MARK(X1)
MARK(2nd(X)) -> MARK(X)
MARK(head(X)) -> MARK(X)
MARK(from(X)) -> MARK(X)
A_{_FROM}(X) -> MARK(X)
MARK(from(X)) -> A_{_FROM}(mark(X))
A_{_SEL}(0, cons(X, XS)) -> MARK(X)
A_{_SEL}(s(N), cons(X, sel(X1', X2'))) -> A_{_SEL}(mark(N), a_{_sel}(mark(X1'), mark(X2')))

Rules:

a_{_from}(X) -> cons(mark(X), from(s(X)))
a_{_from}(X) -> from(X)
a_{_head}(cons(X, XS)) -> mark(X)
a_{_head}(X) -> head(X)
a_{_2nd}(cons(X, XS)) -> a_{_head}(mark(XS))
a_{_2nd}(X) -> 2nd(X)
a_{_take}(0, XS) -> nil
a_{_take}(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a_{_take}(X1, X2) -> take(X1, X2)
a_{_sel}(0, cons(X, XS)) -> mark(X)
a_{_sel}(s(N), cons(X, XS)) -> a_{_sel}(mark(N), mark(XS))
a_{_sel}(X1, X2) -> sel(X1, X2)
mark(from(X)) -> a_{_from}(mark(X))
mark(head(X)) -> a_{_head}(mark(X))
mark(2nd(X)) -> a_{_2nd}(mark(X))
mark(take(X1, X2)) -> a_{_take}(mark(X1), mark(X2))
mark(sel(X1, X2)) -> a_{_sel}(mark(X1), mark(X2))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(0) -> 0
mark(nil) -> nil

The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes